Currently, the bill is set to target four fees: Event/entertainment fees, airline fees, early termination fees for tv/phone/internet services, and surprise resort/destination fees. However, I have heard from several sources that this will instead increase the total cost to consumers as firms will charge more to make up for lost revenue, raising prices until marginal revenue is equal to marginal cost. Would the bill ultimately be beneficial to consumers, or is it only a temporary fix?
You are asking question about effect across various completely different industries with varying industry structures so it is impossible to provide some concrete result, so this answer will only discuss it generally.
What you heard is correct assuming that these companies are perfectly competitive. If a company is perfectly competitive then indeed one can show that price (which is marginal revenue for company) will be equal to marginal cost ($p=MC$) and in such case companies would just translate these costs to their general prices (see discussion in any Microeconomic textbook for example Frank Microeconomics and Behavior). In addition, those fees were there presumably to also set up incentives to discourage behavior this might lead to even higher costs than just having the fees which would further increase the prices.
However, if firms are not perfectly competitive the point about cost increase being reflected in prices is no longer guaranteed. Outside perfect competition $p=MC$ is no longer guaranteed, although it can occur in certain situations (see ibid or Belleflamme & Peitz IO book for some examples). Generally non-competitive firms will have pricing given by some markup that depends on various factors, most importantly price elasticity of demand (Belleflamme & Peitz Industrial Organization ch 3). In such case we have some $p= \mu MC$ with $\mu \geq 1$.
Now as long as $\mu>1$ it is technically possible for company to not raise prices when costs increase as company could in principle simply lower its markup. As was already mentioned markup is a function of price elasticity of demand ($\epsilon_d$). When prices change elasticity usually changes as well (unless we deal with a special case demand function exhibiting constant $\epsilon_d$), so typically any price change would trigger some change in markup. What change in markup will occur is impossible to say without knowing some parameters that have to be estimated, we can just say that higher $\epsilon_d$ means lower markup and vice versa. Also markup could also depend on richer set of variables depending on what model we are dealing with, but elasticity is the major one to take into consideration.
Hence, depending what happens to markup which could go either up or down prices could increase more than marginal cost or less than marginal cost. If markup was sufficiently high and drops now in principle the increase in marginal cost could be even fully offset by drop in markup. It is impossible to say much more without empirical analysis of the effect on some industry.
The best one can do is to say that in highly competitive segments such as regular hotels or entertainment prices will likely rise when marginal costs raise, whereas when it comes to less competitive segments such as some fancy resort that offers unique experience the effect will be ambiguous. Such resort might be forced to absorb the cost increase by lowering its markup if elasticity of demand is high, or might actually increase prices more than marginal cost increase if elasticity of demand is low.